I feel like I have a pretty good way to talk about $\varepsilon$-$\delta$ to physics and engineering students (and possibly students in comp sci). But I am not very sure what I can do for chemistry and biology majors.

For instance, with physics / engineering, I feel like I can talk about designing an apparatus ($f$) whose output is some target value ($L$) with a given tolerance level ($\varepsilon$). How careful do they need to control their input ($\delta$)... etc etc

But when it comes to biology? Or chemistry? I am not sure what a good motivating analogy would be.

EDIT: Cross-posted on MathEducators.SE.

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    $\begingroup$ Might be more suitable for MathEducators.SE $\endgroup$ – Asaf Karagila Feb 3 '15 at 4:00
  • $\begingroup$ It's not as general as the machine example, but "if you need the concentration to be at most $\varepsilon$, how little solute $\delta$ must be in the solution?" might get chemistry students started. You can do a similar analogy for limits at infinity: "if you need the concentration to be at most $\varepsilon$", how large must the volume be, if the mass of solute is given?" Unfortunately both of these problems are linear. $\endgroup$ – Ian Feb 3 '15 at 4:04
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    $\begingroup$ Are you allowed to hit them? $\endgroup$ – Will Jagy Feb 3 '15 at 4:16
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    $\begingroup$ You'll probably find a trove of discussions about whether or not (and how) to teach $\epsilon-\delta$ proofs to students of all backgrounds. I would think any decent introductory calculus book (say, Stewart) which has plenty of pictures and examples is the way to go. $\epsilon-\delta$ concepts are usually something 99% of people, math or otherwise, struggle on the first time. So just patiently present examples and move on to more interesting things as soon as possible (as most introductory calculus books do). $\endgroup$ – Alex R. Feb 3 '15 at 4:31
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    $\begingroup$ This may help (my math teacher in undergrad said it) - 'I come into class on the first day of term and tell you "I don't like students talking in class" and then go about teaching. When I stop and ask you if you've understood there is no response. I smile and tell you that I appreciate questions no matter how trivial they may be and then all of a sudden the class gets more interactive yet not unruly. The variable here is my statement, the function is your reaction and a small change in my statement ($\delta$) gives a small change in your reaction ($\varepsilon$) $\endgroup$ – R_D Feb 3 '15 at 6:16

To explain $f(x) \xrightarrow[x\to a]{} A$ to anyone, I'll choose a graphical approach. In this way, concepts can be visualized. As there're more graphs and diagrams in biology and chemistry, they will be understood better than logic symbols.

  1. Plot the curve $y=f(x)$ (If the limit point $a$ doesn't belong to the domain, leave it out. See the picture below.)
  2. Draw a horizontal strip centered at $y=A$ with height $2\epsilon$. $$ \Large\bbox[5px, #FDF, border: 1px solid black]{ y = A \quad \begin{array}{cc} \hline \updownarrow \epsilon & \phantom{\Rule{6em}{1ex}{0px}} \\ \hdashline \updownarrow \epsilon & \\ \hline \end{array} } \tag{H-strip} \label{fig:hstrip} $$
  3. Find a vertical strip of width $2\delta$ so that the deleted curve (point $a$ removed) won't burst out of the rectangle. $$ \require{extpfeil} \Newextarrow{\lfa}{5,5}{0x2194} \Large\bbox[5px, #FDF, border: 1px solid black]{ y = A \quad \begin{array}{c|c|c|c} & & & \\ \hline \updownarrow \epsilon & {\small \text{curve}} & {\small \text{inside}} & \\ \hdashline \updownarrow \epsilon & {\small \text{these}} & {\small\text{grids}} & \\ \hline & \lfa[\delta]{} & \lfa[\delta]{} & \end{array} } \tag{V-strip} \label{fig:vstrip} $$ If no such $\delta$ can be found, then this $\epsilon$-$\delta$ definition is not satisfied.
  4. Repeat steps (2) and (3) with smaller $\epsilon$. If a (thin) \ref{fig:vstrip} satisfying the criterion in step (3) can be drawn no matter how thin the \ref{fig:hstrip} is, then this $\epsilon$-$\delta$ definition is satisfied.
  5. To consolidate their understanding, ask them to apply these steps to these functions with limit point $a = 0$:
    • $f(x) = kx$, where $k$ is a constant

      $\delta = \frac{\epsilon}{k}$ responds to the $\epsilon$-challenge.

    • $f(x) = \frac{x}{x}$ undefined at $x=0$

      $\delta = \epsilon$ responds to the $\epsilon$-challenge.

    • $f(x) = \frac{\sin x}{x}$ undefined at $x=0$

      $\delta = \epsilon$ responds to the $\epsilon$-challenge.

    • $f(x) = \sin\frac1x$ undefined at $x=0$

      Infinitely many peaks $(((n+\frac12)\pi)^{-1},1)$ and troughs $(((n-\frac12)\pi)^{-1},-1)$ lie out of the four $\epsilon$-$\delta$ grids.

    • $f(x) = x \sin\frac1x$ undefined at $x=0$

      $\delta = \epsilon$ responds to the $\epsilon$-challenge.

Wiki epsilon delta
Picture from Wiki Commons

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